Browsing Mathematics by Issue Date
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On Particle Interaction Models
(20140224)This dissertation deals with three problems in Stochastic Analysis which broadly involve interactions, either between particles (Chapters 1 and 2), or between particles and the boundary of a C2 domain (Chapter 3). In Chapter ... 
On special Lagrangian equations
(20140224)In this paper we study the special Lagrangian equation and related equations. Special Lagrangian equation originates in the special Lagrangian geometry by HarveyLawson [HL1]. In subcritical phases, we construct singular ... 
LamWilliams Markov chains on symmetric groups
(20140224)This paper reviews the current state of the LamWilliams conjectures on a multivariate Markov chain on the symmetric group S_n. We start with Lam's work on random core partitions which led to a remarkable Markov chain on ... 
Combinatorial Laguerre Series
(20140224)We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials. We describe how such a series can be computed by ... 
Toward the compactification of the stack of Lie(G)forms using perfect complexes
To establish geometric properties of an algebraic stack, one can find a compactification. This method has been successfully employed to find irreducible components for example of the moduli stack of curves [DM69], vector ... 
Interacting particle systems with partial annihilation through membranes
This thesis studies the <italic>hydrodynamic limit</italic> and the <italic>fluctuation limit</italic> for a class of interacting particle systems in domains. These systems are introduced to model the transport of positive ... 
The Grothendieck Groups of Module Categories over Coherent Algebras
Let <italic>k</italic> be a field and <italic>B</italic> either a finitely generated free <italic>k</italic>algebra, or a regular <italic>k</italic>algebra of global dimension two with at least three generators, generated ... 
The regularity of Loewner curves
The Loewner differential equation, a classical tool that has attracted recent attention due to SchrammLoewner evolution (SLE), provides a unique way of encoding a simple 2dimensional curve into a continuous 1dimensional ... 
Inverse BoundaryValue Problems on an Infinite Slab
In this work, we study the stability aspect of two inverse boundaryvalue problems (IBVPs) on an infinite slab with partial data. The uniqueness aspects of these IBVPs were considered and studied by Li and Uhlmann for the ... 
Eigenvalue fluctuations for random regular graphs
One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are <italic>universal</italic>. We probe the edges of universality by studying ... 
Arithmetic Properties of the Derived Category for CalabiYau Varieties
This thesis develops a theory of arithmetic FourierMukai transforms in order to obtain results about equivalences between the derived category of CalabiYau varieties over nonalgebraically closed fields. We obtain answers ... 
Selfshrinking Solutions to Mean Curvature Flow
We construct new examples of selfshrinking solutions to mean curvature flow. We first construct an immersed and nonembedded sphere selfshrinker. This result verifies numerical evidence dating back to the 1980's and shows ... 
Shimura Degrees for Elliptic Curves over Number Fields
A crowning achievement of Number theory in the 20th century is a theorem of Wiles which states that for an elliptic curve E over <bold>Q</bold> of conductor N, there is a nonconstant map from the modular curve of level N ... 
Three Problems in Discrete Probability
In this thesis we present three problems. The first problem is to find a good description of the number of fixed points of a 231avoiding permutation. We use a bijection from Dyck paths to 231avoiding permutations that ... 
Timelike graphical models
We study continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a continuous version of graphical ... 
Brownian Motion on Spaces with Varying Dimension
In this thesis we introduce and study Brownian motion with or without drift on state spaces with varying dimension. Starting with a concrete such state space that is the plane with an infinite pole on it, we construct a ... 
Dual Equivalence Graphs and their Applications
In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's ... 
Towards a nonQGorenstein Minimal Model Program
In this thesis we do the first steps towards a nonQGorenstein Minimal Model Program. We extensively study nonQfactorial singularities, using the techniques introduced by [dFH09]. We introduce a new class of singularities, ... 
Path algebras and monomial algebras of finite GKdimension as noncommutative homogeneous coordinate rings
This thesis sets out to understand the categories QGr A where A is either a monomial algebra or a path algebra of finite GelfandKirillov dimension. The principle questions are: \begin{enumerate} \item What is the structure ... 
Grothendieck Duality on Diagrams of Schemes
The Du Bois complex and Du Bois singularities, which extend results of Hodge theory to singular complex varieties, can be defined in terms of a cubical hyperresolution. In this dissertation I further develop the language ...